Earliest Known Uses of Some of the Words of Mathematics (L)

Last revision: July 7, 2017

LAG in time series analysis. In 1901 R. H. Hooker described "a measure of the lag of one phenomenon
behind another upon which it is in some way dependent" in his paper "Correlation of the Marriage-Rate with Trade,"
Journal of the Royal Statistical Society,64, pp. 485-492. David (1998).

LAGRANGE MULTIPLIER.Joseph-Louis
Lagrange states the general
principle for maximising a function of n variables when there are one
or more equations between the variables in his
Théorie des Fonctions Analytiques
(1797, p. 198): "il suffira d'ajouter à la function proposée les functions qui
doivent être nulles, multipliées chacune par une quantité indéterminée ...".
Lagrange originally applied the multiplier technique to problems in the calculus
of variations in his
Mécanique Analytique
(1788, pp. 46-7). (See H. H. Goldstine A History of the Calculus of Variations from the 17th
through the 19th Century (1980).)

Although "Lagrange multiplier" is the standard term today,
"undetermined multiplier" and "indeterminate multiplier"
were the usual terms in the 19th century and for much of the 20th.

The term "Lagrange’s method of
undetermined multipliers" appears in J. W. Mellor, Higher
Mathematics for Students of Chemistry and Physics (1912) [James
A. Landau].

The term "Lagrange multiplier rule" appears in "The Problem of Mayer
with Variable End Points," Gilbert Ames Bliss, Transactions of the
American Mathematical Society, Vol. 19, No. 3. (Jul., 1918).

Lagrange multiplier is found in "Necessary Conditions in the
Problems of Mayer in the Calculus of Variations," Gillie A. Larew,
Transactions of the American Mathematical Society, Vol. 20,
No. 1. (Jan., 1919): "The [lambda]’s appearing in this sum are the
functions of x sometimes called Lagrange multipliers."

The use of Lagrangian for the augmented function
dates from the 1960s, see e.g. Samuel Zahl "A Deformation Method for Quadratic Programming,"
Journal of the Royal Statistical Society, B, 26, (1964), p.
153. (JSTOR search) The Lagrangian function or the Lagrangian expression
were once the popular terms.

Lagrange multiplier test in Statistics.
This test principle was introduced by S. D. Silvey "The
Lagrangian Multiplier Test," Annals of Mathematical Statistics,
30, (1959), 389-407. However, while Silvey’s derivation was new, the
test statistic was already in the literature as the "score test." Econometricians
tend to favour the Lagrange term and statisticians the score term.

Lagrange’s theorem appears in An Elementary Treatise on
Curves, Functions and Forms (1846) by Benjamin Peirce: "The
theorem (650) under this form of application, has been often called
Laplace’s Theorem; but, regarding this change as obvious and
insignificant, we do not hesitate to discard the latter name, and
give the whole honor of the theorem to its true author, Lagrange."

Lagrange’s formula for interpolation appears in 1849 in An
Introduction to the Differential and Integral Calculus, 2nd ed.,
by James Thomson.

Lagrange’s method of approximation occurs in the third edition
of An Elementary Treatise on the Theory of Equations (1875) by
Isaac Todhunter.

LAGRANGIAN (as a noun) occurs in Th. Muir, "Note on the
Lagrangian of a special unit determinant," Transactions Royal Soc.
South Africa (1929).

LAPLACE’S COEFFICIENTS. According to Todhunter (1873),
"the name Laplace’s coefficients appears to have been first used" by
William Whewell (1794-1866) [Chris Linton].

Laplace’s coefficients appears in the title Mathematical
tracts Part I: On Laplace’s coefficients, the figure of the earth,
the motion of a rigid body about its center of gravity, and
precession and nutation (1840) by Matthew O'Brien.

The LAPLACE EXPANSION of a determinant is generally traced to Laplace’s memoir
in Histoire de l'Académie royale des sciences1776 (Année 1772, 2e partie) pp. 267-376
(see pp. 294-304). Thomas Muir makes a detailed examination of the argument in
The Theory of Determinants in the Historical Order of
Development vol. 1, pp. 24-33: he concludes, “there can be no doubt
that if any one name is to be attached to the theorem it should be that of Laplace.” [John Aldrich]

LAPLACE’S FUNCTIONS appears in English in 1833
in Elementary principles of the theories of electricity, heat and molecular actions
by Robert Murphy. [Google print search]

The term LAPLACE’S OPERATOR (for the differential operator
2) was used in 1873 by James Clerk Maxwell in
A Treatise on Electricity and Magnetism
(p. 29): "...an operator occurring in all parts of Physics, which we may
refer to as Laplace’s Operator" (OED2).

The term LAPLACE TRANSFORM was used by Boole and
Poincaré. According to the website of the University of St.
Andrews, Boole and Poincaré might otherwise have used the term
Petzval transform but they were influenced by a student of
Józeph Miksa Petzval (1807-1891) who, after a falling out with
his instructor, claimed incorrectly that Petzval had plagiarised
Laplace’s work.

LAPLACIAN (as a noun, for the differential operator
2)
was used in 1935 by Pauling and Wilson in Introd. Quantum Mech. (OED2).

Latin square appears in English in 1890 in the title of a
paper by Arthur Cayley, "On Latin Squares" in Messenger of Mathematics.
Graeco-Latin square appears in H. F. MacNeish
"Euler Squares." Ann. Math.23, (1921-1922), 221-227.

LATITUDE and LONGITUDE. Henry of Ghent used the
word latitudo in connection with the concept of latitude
of forms.

Nicole Oresme (1320-1382) used the terms latitude and longitude
approximately in the sense of abscissa and ordinate.

LATTICE and LATTICE THEORY in algebra. These terms were introduced by
Garrett Birkhoff in his
“On the combination of subalgebras,” Proceedings of the Cambridge Philosophical Society, 29, (1933), 441–464
and became well-known through his book Lattice Theory (1940). See
Enyclopedia of Mathematics where earlier work by
Schröder and Dedekind and contemporary work by Ore are also described.

The earliest use of lattice point in English given by the OED is from
Cayley’s translation of Eistenstein, which is titled “Eisenstein’s
Geometrical Proof of the Fundamental Theorem for Quadratic Residues” and appears in the
Quarterly Mathematical Journal, vol. I. (1857), pp. 186-191:
“Imagine now in a plane, a rectangular
system of coordinates (x, y) and the whole plane divided by
lines parallel to the axes at distances = 1 from each other into
squares of the dimension = 1. And let the angles which do not lie on
the axes of coordinates be called lattice points.”

[James A. Landau]

The term LATUS RECTUM was used by Gilles Personne de Roberval
(1602-1675) in his lectures on Conic Sections. The lectures were
printed posthumously under the title Propositum locum geometricum
ad aequationem analyticam revocare,... in 1693 [Barnabas Hughes].

Latus rectum is found in English in A Mathematical Dictionary
by Joseph Ralphson: “In a Parabola the Rectangle of the Diameter, and Latus Rectum, is equal to the Rectangle of the Segments of the double Ordinate.” [OED]

LAW OF COSINES is found in 1895 in
Plane and spherical trigonometry, surveying and tables by George Albert Wentworth:
"Law of Cosines. ... The square of any side of a triangle is equal to the sum of the
squares of the other two sides, diminished by twice their product into the cosine of the included angle"
[University of Michigan Digital Library].

The term LAW OF INTERTIA OF QUADRATIC FORMS was introduced by
James Joseph Sylvester
in his “A demonstration of the theorem that every homogeneous
quadratic polynomial is reducible by real orthogonal substitutions to the form
of a sum of positive and negative squares,” Philosophical
Magazine, 4, 138-142. It is sometimes called Sylvester’s
law of inertia. See Kline (p. 799) and the entry in Encyclopedia
of Mathematics. [John Aldrich]

By Poisson’s time there were several theorems that could be covered by this
phrase. The first, given by
Jacob Bernoulli in
Ars Conjectandi (1713), was about "Bernoulli
trials" (a 20th century term). It was often called "Bernoulli’s theorem":
see e.g. Todhunter’s
A History of the Mathematical Theory of Probability
(1865, p. 71). In the course of the 19th century analogous results
were found for other types of random variable.

In the 20th century a new kind of convergence result was obtained,
based on almost-sure convergence, not convergence in probability, as in the Bernoulli tradition. The first of these
results (on Bernoulli trials) was given by E.
Borel in 1909 (see NORMAL NUMBER) and a
more general result was given by
F. Cantelli in 1917. In 1928
A. Y. Khintchine
introduced the term strong law of large numbers to distinguish these results from the
"ordinary" Bernoulli-like results: "Sur la loi forte des grands
nombres," Comptes Rendus de l'Académie
des Sciences,186 page
286.

The obvious term for the Bernoulli results viz., the weak
law of large numbers, seems to have come later. A JSTOR search (restricted
to journals in English) produced
W. Feller’s 1945 "Note on the
Law of Large Numbers and "Fair"
Games," Annals of Mathematical Statistics, 16, 301-304.

In English, law of quadratic reciprocity is found in H. J. Stephen, “Report on the Theory of Numbers.—Part III,”
Report Of The Thirty-First Meeting Of The British Association For The Advancement Of Science;
Held At Manchester In September 1861:
Page 323
“But it follows from the law of quadratic reciprocity, that
one-half of these complete characters are impossible; i. e. that no
quadratic form characterized by them can exist.”
Page 324n
“It is also to be noticed that Gauss does not use the law of
quadratic reciprocity to demonstrate the impossibility of one-half of
the generic characters; for, as we shall hereafter see, this
impossibility is proved in the Disq. Arith. (art. 261) independently of the law of reciprocity, and is then employed to establish that law.”

[James A. Landau]

LAW OF SINES (Snell’s law).The law of sines is found in 1851-54 in Hand-books
of natural philosophy and astronomy by Dionysius Lardner [University
of Michigan Digital Library].

LAW OF SINES (trigonometry) is found in 1895 in
Plane and spherical trigonometry, surveying and tables by George Albert Wentworth:
"...the Law of Sines, which may be thus stated:
The sides of a triangle are proportional to the sines of the opposite angles
[University of Michigan Digital Library].

LAW OF SMALL NUMBERS is a
translation of the German phrase coined by L. von Bortkiewicz, and used by him
as a title of his book Das Gesetz der kleinen Zahlen (1898). (David (2001)).
A JSTOR search found the English phrase in a note, probably by Edgeworth,
in the Economic Journal (1904, p. 496) on an Italian publication
treating “the relation between statistics and the Calculus of Probabilities
with special reference to Prof. Bortschevitch’s ‘law of small numbers.’”

LAW OF TANGENTS is found in 1895 in
Plane and spherical trigonometry, surveying and tables by George Albert Wentworth:
"Hence the Law of Tangents:
The difference of two sides of a triangle is to their sum as the tangent of half the
difference of the opposite angles is to the tangent of half their sum"
[University of Michigan Digital Library].

LAW OF THE ITERATED LOGARITHM is found in English in Philip Hartman and Aurel Wintner, "On
the law of the iterated logarithm," Am. J. Math.63, (1941), 69-176.

LEAST COMMON MULTIPLE.Common denominator appears in
English in 1594 in Exercises by Blundevil: "Multiply the
Denominators the one into the other, and the Product thereof shall
bee a common Denominator to both fractions" (OED2).

Common divisor was used in 1674 by Samuel Jeake in
Arithmetick, published in 1696: "Commensurable, called also
Symmetral, is when the given Numbers have a Common Divisor" (OED2).

Least common multiple is found in 1823 in J. Mitchell,
Dict. Math. & Phys. Sci.: "To find the least common
Multiple of several Numbers" (OED2).

Least common denominator is found in 1844 in Introduction to
The national arithmetic, on the inductive system
by Benjamin Greenleaf: "RULE. - Reduce the fractions, if necessary, to the least common
denominator. Then find the greatest common divisor of the
numerators, which, written over the least common denominator,
will give the greatest common divisor required"
[University of Michigan Digital Library].

Lowest common denominator appears in 1854 in
Arithmetic, oral and written, practically applied by means of
suggestive questions by Thomas H. Palmer:
"Suggestive Questions. - Are all the underlined factors to be
found in the denominators of the fractions marked a and b?
Should they be omitted, then, in finding the lowest common
denominator? What is the product of the factors that are not
underlined? (80·3·5.) Has this product every factor contained
in all the given denominators? Will it form their common
denominator, then? Does it contain no more factors than they
do? Will it form, then, their lowest common denominator?"
[University of Michigan Digital Library].

Least common dividend appears in 1857 in Mathematical
Dictionary and Cyclopedia of Mathematical Science.

Lowest common multiple appears in 1873 in
Test examples in algebra, especially adapted for use in
connection with Olney’s School, or University algebra
by Edward Olney [University of Michigan Digital Library].

LEAST SQUARES. See METHOD OF LEAST SQUARES.

LEBESGUE INTEGRAL. In 1899-1901
Henri Lebesgue
published five short papers in the Comptes Rendus. These formed the basis of his doctoral
dissertation Intégrale, longueur, aire,
published in 1902 in the Annali di
Matematica. The fifth paper of the series “Sur une généralisation de
l’intégral défini,”
Comptes rendus 132, (1901) 1025-1028
announced Lebesgue’s generalisation of the Riemann Integral. (From T. Hawkins
Lebesgue’s Theory of Integration: Its Origins and Development. 1970)

The term Lebesgue integral soon appeared in English, in a paper by
William
H. Young, “On an extension of the Heine-Borel Theorem,” Messenger of Mathematics33
(1903-04), 120-132. The date received by
the editors is not given, but Ivor Grattan-Guinness [“Mathematical
bibliography for W. H. and G. C. Young,” Historia
Mathematica2 (1975),
43-58] places this paper chronologically between papers with received dates of
29 October 1903 and 6 December 1903.

The theorem in
question has, as far as I know, not hitherto been formulated, though it can be
deduced without difficulty from a theorem in a recent memoir by M. Lebesgue,
which states that the Lebesgue integral, as we may conveniently call it, of a
sum of any two functions (as far as our present knowledge of functions goes) is
the sum of their Lebesgue integrals. It has only to be shown that the new
notion of the Lebesgue integral coincides in the case of semi-continuous
functions with the well-known one of upper or lower integral.

(footnote on p. 129)

Behind Young’s reference to the Lebesgue integral is a
tale of lost priority. For when Young next refers to the integral he indicates
the resemblance between Lebesgue’s work and his own researches. William H.
Young, “On upper and lower integration,” Proceedings of the London Mathematical Society (2)
(1905), 52-66. [Received by the editors
on January 14, 1904. The following footnote appears on the first page of the
paper and is dated April 2, 1904.]

This paper was
written simultaneously with the preceding memoir [Young’s "Open sets and
the theory of content"], at a time when the writer was unacquainted with
the work of M. Lebesgue. The result of Theorem 2 is in perfect accord with
Lebesgue’s expression for his integral as the common limit of two difference
summations (Annali di Matematica, 1902, p. 253); in fact, it is easily shown that, in the case of an (upper)
lower semi-continuous function, the Lebesgue integral coincides with the upper
(lower) integral. It may be further remarked that, in the general case, the
Lebesgue integral may itself be expressed in precisely my form.

In accordance with the alterations made in the preceding memoir (cp. footnote, p. 16), I have
made a few verbal alterations in the present paper; I have also elaborated the
proof of the final theorem, which, in its original form, was too condensed.

The following is from p. 143 of Ivor Grattan-Guinness,
“A mathematical union: William Henry and Grace Chisholm Young,”
Annals of Science 29(2) (August 1972),
105-186:

By 1904 Will had, independently of Lebesgue, constructed by different means an equivalent
theory of integration. It was his first really important idea in mathematics
and the discovery that he had been anticipated would have cracked many a lesser
man. But he took it magnanimously; when he heard of Lebesgue’s work he withdrew
his major paper and rewrote parts of it to include considerations on what he
named for ever as 'the Lebesgue integral'. There were enough technical
differences between the two approaches for Young to present his own results in
this paper, and sufficient applications of his approach to all branches of
analysis to make it significant in its time. In fact, some of the succeeding
workers on 'Lebesgue integration' have preferred to follow Young’s rather than
Lebesgue’s approach.

[This entry was contributed by Dave L. Renfro.]

LEG for a side of a right triangle other than the hypotenuse
is found in English in 1659 in Joseph Moxon, Globes (OED2).

Leg is used in the sense of one of the congruent sides of an
isosceles triangle in 1702 Ralphson’s Math. Dict.:
"Isosceles Triangle is a Triangle that has two equal Legs"
(OED2).

LEIBNIZ SERIES. See GREGORY’s SERIES.

LEMMA appears in English in the 1570 translation by Sir Henry
Billingsley of Euclid’s Elements (OED2). [The plural of
lemma can be written lemmas or lemmata.]

LEVERAGE in least squares estimation. The earliest JSTOR appearances are
from 1978 but the term was apparently already established for David F. Andrews and Daryl Pregibon
write that "observations with large effects" are usually called "leverage points".
"Finding the Outliers that Matter," Journal of the Royal Statistical Society. Series
B, 40, (1978), 85-93.

In 1891 in An Elementary Treatise of the Differential and Integral Calculus
by George A. Osborne, the method is not named: "The Differential Calculus furnishes the following method
applicable to all cases."

In Differential and Integral Calculus (1902) by Virgil Snyder and John Irwin Hutchinson, the
procedure is termed "evaluation by differentiation." The same term is
used in Elementary Textbook of the Calculus (1912) by the same
authors.

de l'Hospital’s theorem on indeterminate forms is found in approximately 1904 in the E. R. Hedrick translation of volume I of
A Course in Mathematical Analysis by Edouard Goursat. The translation carries the date 1904, although a footnote
references a work dated 1905 [James A. Landau].

The 1906 edition of A History of Mathematics by Florian Cajori, referring to
L'Hopital’s 1696 treatise, has: "This contains for the first time the method of finding the limiting value of a fraction
whose two terms tend toward zero at the same time."

In Differential and Integral Calculus (1908) by Daniel
A. Murray, the procedure is shown but is not named.

James A. Landau has found in J. W. Mellor, Higher Mathematics for
Students of Chemistry and Physics, 4th ed. (1912), the sentence,
"This is the so-called rule of l'Hopital."

L'Hopital’s rule is dated 1944 in MWCD11.

The rule is named for Guillaume-Francois-Antoine de l'Hospital
(1661-1704), although the rule was discovered by Johann Bernoulli.
The rule and its proof appear in a 1694 letter from him to
l'Hospital.

The family later changed the spelling of the name to l'Hôpital.

LIAR PARADOX. This paradox exists in several forms. One is attributed
to the philosopher Epimenides in the sixth century BC: "All Cretans are
liars...One of their own poets has said so." Another is attributed to Eubulides
of Miletus a leader of the Megarian school from the fourth century BC. In his
Life of Euclides
Diogenes Laertius wrote that Eubulides "handed down a great many arguments in dialectic."
The Eubulides form is "Is the man a liar who says that he tells
lies?" W. & M. Kneale The Development of Logic (1962) pp. 227-8 say
that new variants were devised in the Middle Ages and they speculate that the medieval
logicians may have rediscovered the paradox from considering
St. Paul’s Epistle to Titus
"One of themselves, a prophet of their own, hath said, The Cretans are always liars,
evil wild beasts, lazy gluttons. This witness is true..." Paul evidently did not
see the paradox in this echo of Epimenides.

Lie group appears in English in 1897 in
"Sophus Lie’s Transformation Groups: A Series of Elementary, Expository Articles" by
Edgar Odell Lovett, The American Mathematical Monthly, Vol. 4, No. 10. [JSTOR search]

The term table of mortality appears in A. De Morgan’s
Essay on Probabilities and their Applications to
Life Contingencies and Insurance Offices (1838). The
OED’s earliest quotation for life table is from 1865,
"Every insurance office bases its transactions upon an instrument which is called
a ‘Life Table’" Reader 25 Feb. 213/1. A JSTOR search found the
term in William A. Guy "On the Duration of Life Among the Families of the Peerage
and Baronetage of the United Kingdom," Journal of the Statistical Society
of London, 8, (1845), 69-77.

Formerly, likelihood was a synonym for probability,
as it still is in everyday English. In his paper
"On
the Mathematical Foundations of Theoretical Statistics"
(Phil. Trans. Royal Soc. Ser. A. 222, (1922), p. 326). Fisher made
clear for the first time the distinction between the mathematical properties
of "likelihoods" and "probabilities" (DSB).

The solution of the problems of calculating
from a sample the parameters of the hypothetical population, which we have put
forward in the method of maximum likelihood, consists, then, simply of choosing
such values of these parameters as have the maximum likelihood. Formally, therefore,
it resembles the calculation of the mode of an inverse frequency distribution.
This resemblance is quite superficial: if the scale of measurement of the hypothetical
quantity be altered, the mode must change its position, and can be brought to
have any value, by an appropriate change of scale; but the optimum, as the position
of maximum likelihood may be called, is entirely unchanged by any such transformation.
Likelihood also differs from probability in that it is not a differential element,
and is incapable of being integrated: it is assigned to a particular point of
the range of variation, not to a particular element of it.

Likelihood was first used in a Bayesian context by Harold
Jeffreys in his "Probability and Scientific Method,"
Proceedings of the Royal Society A, 146, (1934) p. 10.
Jeffreys wrote "the theorem of Inverse Probability" in the form

Posterior Probability Prior Probability × Likelihood

This entry was contributed by John Aldrich, based on
David (2001). See BAYES, MAXIMUM LIKELIHOOD, INVERSE PROBABILITY and POSTERIOR & PRIOR.

LIKELIHOOD PRINCIPLE. This expression burst into print in 1962, appearing in "Likelihood
Inference and Time Series" by G. A. Barnard, G. M. Jenkins, C. B. Winsten (Journal of the Royal
Statistical Society A,125, 321-372), "On the Foundations of Statistical Inference" by
A. Birnbaum (Journal of the American Statistical Association,57, 269-306),
and L. J. Savage et al, (1962) The Foundations of Statistical Inference. It must have been current
for some time because the Savage volume records a conference in 1959; the term appears in Savage’s
contribution so the expression may have been his coining.

The LIKELIHOOD RATIO figured in the test theory of J. Neyman and E. S. Pearson from the
beginning, "On the Use of Certain Test Criteria for Purposes of Statistical Inference, Part I" Biometrika,
(1928), 20A, 175-240. They usually referred to it as the likelihood although the phrase "likelihood
ratio" appears incidentally in their "Problem of k Samples," Bulletin Académie Polonaise
des Sciences et Lettres, A, (1931) 460-481. This phrase was more often used by others writing about Neyman
and Pearson’s work, e.g. Brandner "A Test of the Significance of the Difference of the Correlation Coefficients
in Normal Bivariate Samples," Biometrika,25, (1933), 102-109.

The standing of "likelihood ratio" was confirmed by S. S. Wilks’s "The Large-Sample Distribution of the
Likelihood Ratio for Testing Composite Hypotheses," Annals of Mathematical Statistics,9,
(1938), 60-620 [John Aldrich, based on David (2001)].

See the entry WALD TEST.

The term LIMAÇON was coined in 1650 by Gilles Persone de
Roberval (1602-1675) [Encyclopaedia Britannica, article:
"Geometry"].

Limaçon is found in English in 1852 in
A Treatise on Higher Plane Curves by G. Salmon:
“The origin is a double point, and the curve is of the fourth
class, and is the ‘limaçon de Pascal.’” [OED]

The curve is sometimes called Pascal’s limaçon,
for Étienne Pascal (1588?-1651), the first person to study it.
Boyer (page 395) writes that "on the suggestion of Roberval" the
curve is named for Pascal.

LIMIT.Gregory
of St. Vincent (1584-1667) used terminus to mean the limit of a progression,
according to Carl B. Boyer in The History of the Calculus and its Conceptual
Development.

Isaac
Newton wrote justifying limits in the Scholium to
Section
I of Book I of the Principia
(Philosophiae Naturalis Principia Mathematica
or The Mathematical Principles of Natural Philosophy) (first edition 1687)

Perhaps it may be objected, that there is no ultimate proportion,
of evanescent qualities; because the proportion, before the quantities have
vanished, is not the ultimate, and when they are vanished, is none. But by the
same argument, it may be alledged, that a body arriving at a certain place,
and there stopping, has no ultimate velocity: because the velocity, before the
body comes to the place, is not its ultimate, velocity; when it has arrived,
is none. But the answer is easy; for by the ultimate velocity is meant that
with which the body is moved, neither before it arrives at its last place and
the motion ceases, nor after, but at the very instant it arrives; that is, that
velocity with which the body arrives at its last place, and with which the motion
ceases. And in like manner, by the ultimate ratio of evanescent quantities is
to be understood the ratio of the quantities not before they vanish, nor afterwards,
but with which they vanish. In like manner the first ratio of quantities is
that with which they begin to be. And the first or last sum is that with which
they begin and cease to be (or to be augmented or diminished). There is a limit
which the velocity at the end of the motion may attain, but not exceed. This
is the ultimate velocity. And there is the like limit in all quantities and
proportions that begin and cease to be. And since such limits are certain and
definite, to determine the same is a problem strictly geometrical. But whatever
is geometrical we may be allowed to use in determining and demonstrating any
other thing that is likewise geometrical. (Translated
by Andrew Motte 1729)

Katz (p. 471) comments, "A translation
of Newton’s words into an algebraic statement would give a definition of limit
close to, but not identical with, the modern one."

In 1821 Augustin-Louis
Cauchy defined limit as follows: "If the successive values attributed to
the same variables approach indefinitely a fixed value, such that they finally
differ from it by as little as one wishes, this latter is called the limit of
all the others." Cours d'analyse
(Oeuvres
II.3), p. 19. (Translation from Katz page 641) Cauchy introduced the modern
ε, δ way of arguing.

Limit point is found in English in E. H. Moore
“A Simple Proof of the Fundamental Cauchy-Goursat Theorem,”
Transactions of the American Mathematical Society, Vol. 1, No. 4. (Oct., 1900), pp. 499-506.

Point of accumulation appears in English in in E. W. Chittenden, “On the
classification of points of accumulation in the theory of abstract
sets,” Bulletin A. M. S. 32 (1926).

LINDLEY’S PARADOX. "An example is produced to show that, if H is a simple hypothesis and x
the result of an experiment, the following two phenomena can occur
simultaneously: (i) a significance test for H reveals that x is significant at, say, the 5% level; (ii) the posterior
probability of H given x, is, for quite small prior probabilities of H, as high as 95%."
D. V. Lindley "A Statistical Paradox"
Biometrika, 44, (1957), pp. 187-192. Lindley notes that "the paradox is not
in essentials new, although few statisticians are aware of it." (p. 190) His earliest
reference is to the discussion in Jeffreys’s Theory of Probability on which he comments, "Jeffreys is concerned to
emphasise the similarity between his tests and those due to Fisher and the
discrepancies are not emphasised." Lindley’s Paradox is the common name although some writers refer to the
Jeffreys-Lindley Paradox.

LINE FUNCTION was the term used for functional by Vito
Volterra (1860-1940), according to the DSB.

LINE GRAPH is found in the Danville Bee on Sept. 8, 1923:
“A line graph shows the number of farms on which each of the crops is grown.”

The term LINE INTEGRAL was used in 1873 by James Clerk Maxwell in a
Treatise on Electricity and Magnetism,
p. 71 in the phrase "Line-Integral of Electric Force, or Electromotive
Force along an Art of a Curve" (OED2). Earlier in the book (p. 12) Maxwell
explained how "Line-integration [is] appropriate to
forces, surface-integration to fluxes." The concept of a line integral
is much older: see the entry GREEN’S THEOREM.

The alternate term curve integral has been seen in
1965 textbook, but it may be much older.

The term LINE OF EQUAL POWER was coined by Steiner.

LINEAR ALGEBRA. The DSB seems to imply that the term
algebra linearia is used by Rafael Bombelli (1526-1572) in
Book IV of his Algebra to refer to the application of
geometrical methods to algebra.

Linear algebra is found in English in 1870 in
the American Journal of Mathematics (1881)4/107:
“An algebra in which every expression is reducible to
the form of an algebraic sum of terms, each of which consists of a single
letter with a quantitative coefficient, is called a linear algebra.”
[OED]

Linear algebra occurs in 1875 in the title, "On the uses and
transformations of linear algebra" by Benjamin Peirce, published in
American Acad. Proc. 2 [James A. Landau].

Pierce meant what today we would call a "finite dimensional
algebra over a field," not the theory of vector spaces and linear
transformations. [Fernando Q. Gouvea]

In 1945, The Development of Mathematics, 2nd. ed. by E. T.
Bell has: “ The introduction of general methods into linear
algebra, beginning in the first decade of the twentieth century,
prepared that vast field of mathematics..for partial arithmetization in
the second and third decades.” [OED]

LINEAR COMBINATION occurs in "On the Extension of
Delaunay’s Method in the Lunar Theory to the General
Problem of Planetary Motion," G. W. Hill, Transactions
of the American Mathematical Society, Vol. 1, No. 2.
(Apr., 1900).

LINEAR DEPENDENCE appears in the title
“The Theory of Linear Dependence” by Maxime Bôcher
published in 1900 in the Annals of Mathematics
[James A. Landau].

LINEAR EQUATION appears in English the 1816 translation of
Lacroix’s Differential and Integral Calculus (OED2).

LINEAR FUNCTION is found in 1843 in
"Chapters in the Analytical Geometry of (n) Dimensions"
by Arthur Cayley in the Cambridge Mathematical Journal, vol. IV
[University of Michigan Digital Library].

Linear function is found in English in volume I of An Elementary Treatise on Curves,
Functions and Forces by Benjamin Peirce. The title page of this work has 1852;
the copyright date on the reverse of the title page is 1841 [James A. Landau].

LINEAR INDEPENDENCE is found in 1901 in Linear Groups, with
an exposition of the Galois field theory by Leonard Eugene
Dickson [James A. Landau].

LINEAR OPERATOR.Linear operation appears in 1837 in
Robert Murphy, "First Memoir on the Theory of Analytic Operations,"
Philosophical Transactions of the Royal Society of London,127, 179-210.
Murphy used "linear operation" in the sense of the modern term "linear operator"
[Robert Emmett Bradley].

LINEAR PRODUCT. This term was used by Hermann Grassman in his
Ausdehnungslehre (1844).

LINEAR PROGRAMMING. See PROGRAMMING.

LINEAR TRANSFORMATION appears in 1843 in the title
“Exposition of a general theory of linear transformations, Part II”
by George Boole in Camb. Math. Jour. t. III. 1843, pp. 1-20. [James A. Landau]

LINEARLY DEPENDENT was used in 1893 in "A Doubly Infinite
System of Simple Groups" by Eliakim Hastings Moore. The paper was
read in 1893 and published in 1896 [James A. Landau].

LINEARLY INDEPENDENT is found in 1847 in "On the Theory of Involution in
Geometry" by Arthur Cayley in the Cambridge and Dublin Mathematical Journal
[University of Michigan Historical Math Collection].

LINK FUNCTION. One of the components of a GENERALIZED LINEAR MODEL
is "the linking function, θ = f(Y) connecting the parameter
of the distribution of z [the
dependent variable] with the Y’s
of the linear model." Nelder & Wedderburn Journal
of the Royal Statistical Society, A, 135, (1972), p. 372.
The term link function was introduced in J. A. Nelder’s "Log Linear Models for Contingency Tables: A Generalization of Classical
Least Squares," Applied Statistics, 23, (1974),
pp. 323-329. (Based on David (1998))

The term LITUUS (Latin for the curved wand used by
the Roman pagan priests known as augurs) was chosen by Roger Cotes
(1682-1716) for the locus of a point moving such that the area of a
circular sector remains constant, and it appears in his Harmonia
Mensurarum, published posthumously in Cambridge, 1722 [Julio
González Cabillón].

The term LOCAL PROBABILITY is due to Morgan W. Crofton
(1826-1915) (Cajori 1919, page 379).

The term is found in 1865 in James Joseph Sylvester,
“On a Special Class of Questions on the Theory of Probabilities,” Birmingham
British Association Report 35:
“After referring to the nature of geometrical or local probability in general,
the author of the paper drew attention to a particular class of questions
partaking of that character in which the condition whose probability is to be ascertained
is one of pure form. The chance of three points within a circle or sphere being apices
of an acute or obtuse-angled triangle, or of the quadrilateral formed by joining four
convex quadrilateral, will serve as types of the class of questions in view.”

The term appears in “Note on Local Probability” by Crofton in
Mathematical Questions, With Their Solutions, Vol. 7, from January to July 1867:
“When, therefore, in questions on Local Probability, we speak of a point
or line taken at random, i.e. according to no law, there is no obscurity
in the idea; but it cannot be made a matter of arithmetical calculation till we have a definite
conception of the nature of the assemblage of points or lines one (or more) of which we take.”
[Herb Acree]

LOCATION and SCALE. The location and scaling
of frequency curves is discussed in §9 of R. A. Fisher’s
"On the Mathematical Foundations of Theoretical
Statistics" (Phil. Trans. R. Soc. 1922, p. 338). In §9 of
Two New Properties of Mathematical Likelihood
(Proc.R. Soc., A, 1934 p. 303) Fisher changed his terminology to the
estimation of location and scaling. The terms
location parameter and scale parameter were used by E. J. G. Pitman
in "Tests of Hypotheses Concerning Location and Scale Parameters," Biometrika,
31, (1939), 200-215.

David (2001)

LOCUS is a Latin translation of the Greek word topos.
Both words mean "place."

According to Pappus, Aristaeus (c. 370 to c. 300 BC) wrote a work
called On Solid Loci (Topwn sterewn).

Pappus also mentions Euclid in connection with locus problems.

Apollonius mentioned the "locus for three and four lines" ("...ton
epi treis kai tessaras grammas topon...") in the extant letter
opening Book I of the Conica. Apollonius said in the first
book that the third book contains propositions (III.54-56) relevant
to the 3 and 4 line locus problem (and, since these propositions are
new, Apollonius claimed Euclid could not have solved the problem
completely--a claim that caused Pappus to call Apollonius a braggard
(alazonikos). In Book III itself there is no mention of the locus
problem [Michael N. Fried].

Locus appears in the title of a 1636 paper by Fermat, "Ad
Locos Planos et Solidos Isagoge" ("Introduction to Plane and Solid
Loci").

In English, locus is found in 1727-41 in Chambers
Cyclopedia: "A locus is a line, any point of which may
equally solve an indeterminate problem. ... All loci of the
second degree are conic sections" (OED2).

Locus geometricus is an entry in the 1771 Encyclopaedia
Britannica.

LOGARITHM. Before he coined the term logarithmus Napier called these
numbers numeri artificiales, and the arguments of his logarithmic function
were numeri naturales [Heinz Lueneburg].

Logarithmus was coined (in Latin) by John
Napier (1550-1617) and appears in 1614 in his Mirifici
Logarithmorum Canonis descriptio.

According to the OED2, "Napier does not explain his view of the
literal meaning of logarithmus. It is commonly taken to mean
'ratio-number', and as thus interpreted it is not inappropriate,
though its fitness is not obvious without explanation. Perhaps,
however, Napier may have used logos merely in the sense of
'reckoning', 'calculation.'"

According to Briggs in Arithmetica logarithmica (1624), Napier
used the term because logarithms exhibit numbers which preserve
always the same ratio to one another.

According to Hacker (1970):

It undoubtedly was Napier’s observation that logarithms
of proportionals are "equidifferent" that led him to coin the name
"logarithm," which occurs throughout the Descriptio but only
in the title of the Constructio, which clearly was drafted
first although published later. The many-meaning Greek word
logos is therefore used in the sense of ratio. But
there is an amusing play on words to which we might call attention
since it does not seem to have been noticed. It is interesting that
the Greeks also employed logos to distinguish
reckoning, or that is to say mere calculation, from
arithmos, which was generally reserved by them to indicate the
use of number in the higher context of what today we call the
theory of numbers. Napier’s "logarithms" have indeed served both
purposes.

Logarithm appears in English in a letter of March 10, 1615,
from Henry Briggs to James Ussher: "Napper, Lord of Markinston, hath
set my Head and Hands a Work, with his new and admirable Logarithms.
I hope to see him this summer, if it please God, for I never saw a
book which pleased me better or made me more wonder."

Logarithm appears in English in 1616 in E. Wright’s English
translation of the Descriptio: "This new course of Logarithmes
doth cleane take away all the difficultye that heretofore hath beene
in mathematicall calculations. [...] The Logarithmes of proportionall
numbers are equally differing."

In the Constructio, which was drafted before the
Descriptio, the term "artificial number" is used, rather than
"logarithm." Napier adopted the term logarithmus before his
discovery was announced.

Jobst Bürgi called the logarithm Die Rothe Zahl since the
logarithms were printed in red and the antilogarithms in black in his
Progress Tabulen, published in 1620 but conceived some years earlier
(Smith vol. 2, page 523).

[Older English-language dictionaries pronounce logarithm with
an unvoiced th, as in thick and arithmetic.]

See also BRIGGSIAN LOGARITHM, COMMON LOGARITHM, NAPIERIAN LOGARITHM, NATURAL LOGARITHM.

Christiaan Huygens used logarithmica when he wrote in Latin and logarithmique
when he wrote in French.

Johann Bernoulli used a phrase which is translated "logarithmic
curve" in 1691/92 in Opera omnia (Struik, page 328).

Logarithmic curve is found in English in 1715
in The Elements of Astronomy, Physical and Geometrical
by David Gregory and Edmond Halley:
"But this is the Property of the Logarithmic Curve very well known to
Geometricians; therefore the Curve ACX is a Logarithmic Curve,
whose Asymptote is the Right line BZ." [Google print search]

Logarithmic function appears in 1831 in the second edition of
Elements of the Differential Calculus (1836) by John Radford
Young: "Thus, ax, a log x, sin
x, &c., are transcendental functions: the first is an
exponential function, the second a logarithmic
function, and the third a circular function" [James A. Landau]

The term LOGARITHMIC POTENTIAL was coined by Carl Gottfried
Neumann (1832-1925) (DSB).

The term LOGARITHMIC SPIRAL was introduced by Pierre Varignon
(1654-1722) in a paper he presented to the Paris Academy in 1704 and
published in 1722 (Cajori 1919, page 156).

Another term for this curve is equiangular spiral.

Jakob Bernoulli called the curve spira mirabilis (marvelous
spiral).

LOGIC. The term logikê (knowledge of the functions of logos or reason)
was used by the Stoics but it covered many philosophical topics that
are not part of the modern subject. According to W. & M. Kneale The Development of Logic
(1962) pp. 7 & 23, the word
"logic" first appeared in its modern sense in the commentaries of
Alexander of Aphrodisias who wrote in the
third century AD. Until the late 19th century the scope of the study
was determined by the contents of
Aristotle’s
(384-322 BC) writings on reasoning.
These were assembled by his pupils after his death and became collectively known
as the Organon or instrument of science. See Robin Smith’s
Aristotle’s Logic.

In the Middle Ages logic was one of the three sciences composing
the ‘trivium’, the former of the two divisions of the seven ‘liberal arts’.
The other constituents were rhetoric and grammar. The higher division, the ‘quadrivium’
consisted of arithmetic, geometry, astronomy and music. For the English word
logic the OED’s earliest quotation is from 1362, the second from
the Prologue to Chaucer’s Canterbury Tales: "A Clerk ther was of
Oxenford also, That unto logik hadde longe ygo." (c.1386.) See QUADRIVIUM.

Although the "Bibliography of Symbolic Logic" published in the first volume of the
Journal of Symbolic Logic (December 1936, pp. 121-216) starts in 1666 with Leibniz, the modern era in
logic begins in the 19th century with the work of
Augustus de Morgan (1806-1871) and
George Boole (1815-1864).
By the end of the century many new terms had been coined, including names for the subject, for
the author’s particular take on it and for its various sub-divisions. Some of
the names are still in use, although their meaning has often shifted. Some authors
kept the new terms distinct, others would use them indifferently, e.g. Bertrand
Russell treated "symbolic logic" and "mathematical logic"
as interchangeable in his "Mathematical Logic as Based on
the Theory of Types," American Journal of Mathematics, 30,
(1908), 222-262.

Formal logic and symbolic logic were
used as book-titles by De Morgan (1847) and J. Venn (1881) respectively. G.
Peano used mathematical logic as the name of his new subject. Its concerns
were not those of traditional logic, as he explained to Felix Klein in 1894:
"the aim of Mathematical logic is to analyse the ideas and reasoning which
feature especially in the mathematical sciences." (quoted on p. 243 of
Grattan-Guinness (2000). E. Schröder used the phrase algebra of logic
in the title of his main work, Vorlesungen über die Algebra der Logik (volume
1, 1890). Logistic (French logistique) was used by Couturat and
other speakers at the International Congress of Philosophy in 1904 and was popular
for a few decades. The terms deductive logic and inductive logic
originated in the 19th century: W. S. Jevons called one of his books
Studies in Deductive Logic (1880) and Venn one of his, The Principles
of Empirical or Inductive Logic (1889). Informal logic is a new term,
having been in use only since the 1970s; see Leo Groarke’s
Informal logic.

This entry was contributed by John Aldrich.
See MATHEMATICAL LOGIC. A complete list of the set theory and logic terms on this web site is
here.
For the symbols of logic see Earliest Use of Symbols.

LOGICISM is the doctrine that mathematics is in some significant sense reducible to logic.
It is associated with the
Principia Mathematica(1910-1913) of
A. N. Whitehead and Bertrand Russell. According to Grattan-Guinness (2000, pp.
479 & 501), the word Logizismus was introduced by A. A. H. Fraenkel
Einleitung in der Mengenlehre (1928) and R. Carnap Abriss der Logistik
(1929). A JSTOR search found the English word in H. Reichenbach "Logical Empiricism in Germany and the Present State
of its Problems," Journal of Philosophy, 33, (1936), p. 143.

This entry was contributed by John Aldrich.
See also FORMALISM and INTUITIONISM.

The term LOGISTIC CURVE is attributed to Edward Wright (ca.
1558-1615) (Thompson 1992, page 145), although Wright used the term to refer to the logarithmic curve.

The term logistic regression appears in D.
R. Cox "The Regression Analysis of Binary Sequences," Journal of the Royal
Statistical Society. Series B (Methodological), 20, (1958), 215-242.

LOGIT first appeared in Joseph Berkson’s "Application to the Logistic Function to Bio-Assay,"
Journal of the American Statistical Association, 39, (1944), p. 361: "Instead of the observations
qi we deal
with their logits li = ln(pi
/ qi). [Note]
I use this term for ln p/q following Bliss, who called the analogous
function which is linear on x for the normal curve ‘probit’." (OED)

See PROBIT.

LOGNORMAL.Logarithmic-normal was used in 1919 by S. Nydell in "The
Mean Errors of the Characteristics in Logarithmic-Normal Distributions,"
Skandinavisk Aktuarietidskrift, 2, 134-144 (David, 1995).

Lognormal was used by J. H. Gaddun in Nature on Oct. 20, 1945: "It
is proposed to call the distribution of x 'lognormal' when the distribution
of log x is normal" (OED2).

The lognormal distribution was apparently first studied when
Donald McAlister answered a question put by Francis Galton: to what "law
of error" does the geometric mean bear the relationship that the arithmetic
mean bears to the normal distribution? See Galton’s
The
Geometric Mean, in Vital and Social StatisticsProceedings of the Royal Society of London, 29, (1879), 365-367,
which prefaced McAlister’s paper
"The
Law of the Geometric Mean." They
did not give a name to the new distribution. However, according to E. T. Whittaker
& G. Robinson Calculus of Observations (1924, p. 218), Seidel had
asked and answered the same question in 1863.

See also ARITHMETIC MEAN and GEOMETRIC MEAN.

LONG DIVISION is found in 1744 in The Schoolmaster’s Assistant
by Thomas Dilworth: “It very seldom happens that the Divisor consists
of more than one Denomination; yet because such Divisors may sometimes offer themselves,
I will give a few for the reader’s satisfaction, which musst be wrought
after the manner of Long Division, and may serve also as proofs to some of the foregoing
examples in Multiplication.” [OED]

LORENZ ATTRACTOR. This object was first described by the
meteorologist Edward Norton
Lorenz in his paper “Deterministic non-periodic flow,”
J. Atmos. Sci.,20 : 2 (1963) pp. 130–141. The term
“Lorenz attractor” came into use in the 1970s when his work
began to be noticed. See MathWorld
and the Encyclopedia of
Mathematics.

LORENZ CURVE. This
diagram was introduced by
Max O. Lorenz in his “Methods of Measuring the Concentration of
Wealth,” Publications of the American
Statistical Association, 9,
(Jun., 1905), pp. 209-219. The term Lorenz
curve quickly entered circulation—see e.g. W. M. Persons “The
Measurement of Concentration of Wealth,” Quarterly
Journal of Economics, 24,
(Nov., 1909), p. 172. According to M. J. Bowman “A
Graphical Analysis of Personal Income Distribution in the United States,” American Economic Review, 35, (1945), p. 617n, “The same idea was
introduced almost simultaneously by Gini, Chatelain and Séailles.” The Séailles
reference is to his 1910 book La répartition
des fortunes en France.

LOSS and LOSS FUNCTION in statistical
decision theory. In the paper establishing the subject ("Contributions to the
Theory of Statistical Estimation and Testing Hypotheses," Annals of Mathematical
Statistics,10, 299-326) Wald referred to "loss" but used
"weight function" for the (modern) loss function. He continued
to use weight function, for instance in his book Statistical Decision Functions
(1950), while others adopted loss function. Arrow, Blackwell & Girshick’s
"Bayes and Minimax Solutions of Sequential Decision Problems" (Econometrica,
17, (1949) 213-244) wrote L rather than W for the
function and called it the loss function. A paper by Hodges & Lehmann ("Some
Problems in Minimax Point Estimation," Annals of Mathematical Statistics,21, (1950), 182-197) used loss function more freely but retained Wald’s
W.

This entry was contributed by John Aldrich, based on David (2001) and JSTOR.
See DECISION THEORY.

The term LOWER SEMICONTINUITY was used by René-Louis Baire (1874-1932), according
to Kramer (p. 575), who implies he coined the term. The term appears in Baire’s thesis,
“Sur les fonctions de variables réelles,” Annali di Matematica Pura ed Applicata (3) 3 (1899), 1-123,
and in his “Sur la théorie des fonctions discontinues,
Comptes rendus,129, (1899) 1010-1013.

The phrase LOWEST TERMS appears in about 1675 in Cocker’s
Arithmetic, written by Edward Cocker (1631-1676): "Reduce a
fraction to its lowest terms at the first Work" (OED2). (There is
some dispute about whether Cocker in fact was the author of the
work.)

LOXODROME. Pedro Nunez (Pedro Nonius) (1492-1577) announced
his discovery and analysis of the curve in De arte navigandi.
He called the curve the rumbus (Catholic Encyclopedia).

The term loxodrome is due to Willebrord Snell van Roijen
(1581-1626) and was coined in 1624 (Smith and DSB, article: "Nunez
Salaciense).

LUCAS-LEHMER TEST occurs in the title, "The Lucas-Lehmer test for Mersenne
numbers," by S. Kravitz in the Fibonacci
Quarterly 8, 1-3 (1970). The test is named for
Edouard Lucas and
Dick Lehmer.

The term Lucas’s test was used in 1932 by A. E. Western in "On
Lucas’s and Pepin’s tests for the primeness of Mersenne’s numbers,"
J. London Math. Soc. 7 (1932), and in 1935 by D. H. Lehmer
in "On Lucas’s test for the primality of Mersenne’s numbers," J.
London Math. Soc. 10 (1935).

The term LUCAS PSEUDOPRIME occurs in the title "Lucas
Pseudoprimes" by Robert Baillie and Samuel S. Wagstaff Jr. in
Math. Comput. 35, 1391-1417 (1980): "If n is composite, but
(1) still holds, then we call n a Lucas pseudoprime with parameters P
and Q ..." [Paul Pollack].

LUDOLPHIAN NUMBER. The number 3.14159... was often called the
Ludolphische Zahl in Germany, for
Ludolph
van Ceulen.

In English, Ludolphian number is found in 1886 in G. S. Carr,
Synopsis Pure & Applied Math (OED2).

In English, Ludolph’s number is found in 1894 in History
of Mathematics by Florian Cajori (OED2).

LUNE.Lunula appears in A Geometricall Practise named Pantometria by Thomas Digges
(1571): "Ye last figure called a Lunula" (OED2).

Lune appears in English in 1704 in Lexicon technicum, or an
universal English dictionary of arts and sciences by John Harris
(OED2).